Dirac-Fock Hamiltonian

Fully relativistic calculations even for atoms are quite complicated. The relativistic ECP parameters are, therefore, usually derived from atomic calculations that include only the most important relativistic terms of the Dirac-Fock Hamiltonian, namely, the mass-velocity correction, the spin-orbit coupling, and the so-called Darwin term.6 This is why the reference atomic calculations and the derived ECP parameters are sometimes termed quasi-relativistic. The basic assumption of relativistic ECPs is that the relativistic effects can be incorporated into the atom via the derived ECP parameters as a constant, which does not change during formation of the molecule. Experience shows that this assumption is justified for calculating geometries and bond energies of molecules. 

The Dirac-Fock Hamiltonian for a many-electron atom can be written as... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

The first two-step calculations of the P,T-odd spin-rotational Hamiltonian parameters were performed for the PbF radical about 20 years ago [26, 27], with a semiempirical accounting for the spin-orbit interaction. Before, only nonrelativistic SCF calculation of the TIF molecule using the relativistic scaling was carried out [86, 87] here the P,T-odd values were underestimated by almost a factor of three as compared to the later relativistic Dirac-Fock calculations. The latter were first performed only in 1997 by Laerdahl et al. [88] and by Parpia [89]. The next two-step calculation, for PbF and HgF molecules [90], was carried out with the spin-orbit RECP part taken into account using the method suggested in [91]. 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

The method discussed here for the inclusion of relativistic effects in molecular electronic structure calculations is grounded in the Dirac-Fock approximation for atomic wave functions (29). The premise is that the major relativistic effects of the Dirac Hamiltonian are manifested in the core region, involving the core electrons, and that these effects propagate to the valence electrons. In addition, there are direct relativistic effects on valence electrons penetrating into the core region. Insofar as this is true, the valence electrons can be treated using a nonrelativistic Hamiltonian to which is added an operator, the relativistic effective core potential (REP). The REP formally, incorporates relativistic effects due to core electrons and to interactions of valence electrons with core electrons in an internally consistent way. 

The Hartree—Fock problem with the Dirac Hamiltonian (3.153) is called Dirac—Fock. The coordinate—spin representation of the orbital rj) is... 

For the relativistic Hamiltonian the procedure is called multiconfiguration Dirac—Fock. A computer program for structure calculations in this approximation has been described by Grant et al. (1980). The non-relativistic procedure has been described by Froese-Fischer (1977) and implemented by the same author (Froese-Fischer, 1978). 

The DV-DFS molecular orbital(MO) method is based on the Dirac-Fock-Slater approximation. This method provides a powerful tool for the study of the electronic structures of molecules containing heavy elements such as uranium[7,8,9,10]. The one-electron molecular Hamiltonian in the Dirac-Fock-Slater MO method is written as... 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

A proper definition of (quasi-)particle-creation and (quasi-)particle-annihilation operators an and a is provided by diagonalization of the (time-independent) unperturbed part Ho = //ext+Z/e-e of the total Hamiltonian. After the iteration is performed (e.g. on the Dirac-Fock level) the latter may be cast into the form... 

Shape-consistent pseudopotentials including spin-orbit operators based on Dirac-Hartree-Fock AE calculations using the Dirac-Coulomb Hamiltonian have been generated by Christiansen, Ermler and coworkers [161-170]. The potentials and corresponding valence basis sets are also available on the internet under http //www.clarkson.edu/ pac/reps.html. A similar, quite popular set for main group and transition elements based on scalar-relativistic Cowan-Griffin AE calculations was published by Hay and Wadt [171-175]. 

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